Math Problem Statement

Solve the following inequalities: 1) x^2 + 8x - 9 >= 0, 2) -x^2 + 5x - 6 < 0, 3) x^2 + x - 2 <= 0

Solution

The image you uploaded contains a set of inequalities to solve. Let me break down each inequality for you:

Inequalities to Solve:

  1. x2+8x90x^2 + 8x - 9 \geq 0
  2. x2+5x6<0-x^2 + 5x - 6 < 0
  3. x2+x20x^2 + x - 2 \leq 0

I will now proceed with solving each of these inequalities step by step.

1. Solve x2+8x90x^2 + 8x - 9 \geq 0:

  • First, we factor the quadratic expression: x2+8x9=(x+9)(x1)x^2 + 8x - 9 = (x + 9)(x - 1)
  • Now, solve the inequality (x+9)(x1)0(x + 9)(x - 1) \geq 0 using the sign chart method. The critical points are x=9x = -9 and x=1x = 1. Analyze the sign of the product in each interval: (,9)(-\infty, -9), (9,1)(-9, 1), and (1,)(1, \infty).

2. Solve x2+5x6<0-x^2 + 5x - 6 < 0:

  • Rewrite this inequality as x25x+6>0x^2 - 5x + 6 > 0 (by multiplying both sides by -1, reversing the inequality).
  • Factor the quadratic expression: x25x+6=(x2)(x3)x^2 - 5x + 6 = (x - 2)(x - 3)
  • Now, solve the inequality (x2)(x3)>0(x - 2)(x - 3) > 0 using the sign chart method. The critical points are x=2x = 2 and x=3x = 3. Analyze the sign of the product in each interval: (,2)(-\infty, 2), (2,3)(2, 3), and (3,)(3, \infty).

3. Solve x2+x20x^2 + x - 2 \leq 0:

  • Factor the quadratic expression: x2+x2=(x+2)(x1)x^2 + x - 2 = (x + 2)(x - 1)
  • Solve the inequality (x+2)(x1)0(x + 2)(x - 1) \leq 0 using the sign chart method. The critical points are x=2x = -2 and x=1x = 1. Analyze the sign of the product in each interval: (,2)(-\infty, -2), (2,1)(-2, 1), and (1,)(1, \infty).

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Inequalities
Sign Chart Method
Factorization

Formulas

x^2 + bx + c = 0 (for factoring quadratics)
(x + a)(x - b) = 0 (for solving quadratic inequalities)

Theorems

Sign Chart Method for solving inequalities
Factorization of Quadratic Equations

Suitable Grade Level

Grades 8-10